Similar characterizations are known for continuous isometric
actions \(\alpha \colon  G \rightarrow \mathop {{\rm Iso}}\nolimits (E)\) 
of a locally compact abelian group \(G\)  on a Banach space \(E\) .
One associates to such a representation its Arveson spectrum
\(\mathop {{\rm Sp}}\nolimits (\alpha ) \subseteq \widehat{G}\)  and then the norm continuity
of \(\alpha \)  is equivalent to the compactness of its
spectrum ([1]}).

